The connection between mathematics and art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius bands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.

Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.

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Hilbert's Square-Filling Curve"Hilbert's Square-Filling Curve" by The
3DXM Consortium

In 1890 David Hilbert published a construction of a continuous curve whose image completely fills a square, which was a significant contribution to the understanding of continuity. Although it might be considered to be a pathological example, today, Hilbert's curve has become well-known for a very different reason---every computer science student learns about it because the algorithm has proved useful in image compression. See more fractal curves on the 3D-XplorMath Gallery.

--- adapted from "About Hilbert's Square Filling Curve" by Hermann Karcher

Mandelbrot SetA striking aspect of this image is its self-similarity: Parts of the set look very similar to larger parts of the set, or to the entire set itself. The boundary of the Mandelbrot Set is an example of a fractal, a name derived from the fact that the dimensions of such sets need not be integers like two or three, but can be fractions like 4/3. See more at the 3D-XplorMath Fractal Gallery.

--- Richard Palais (Univ. of California at Irvine, Irvine, CA)

Hopf Fibered Linked Tori"Hopf Fibered Linked Tori," by The
3DXM Consortium

The Hopf map maps the unit sphere in four-dimensional space to the unit sphere in three-dimensional space. The four tori linked in this image are made up of fibers, or pre-images, of the Hopf map. In this visualization, each fiber has a constant color and the color varies with the distance of the fibers. Any two of the four tori are linked, as are any pair of fibers on a given torus. See more surface images on the 3D-XplorMath Gallery.

--- adapted from "Hopf Fibration and Clifford Translation of the 3-Sphere," by Hermann Karcher

Parametric Breather"Parametric Breather," by The 3DXM Consortium.

This striking object is an example of a surface in 3-space whose intrinsic geometry is the hyperbolic geometry of Bolyai and Lobachevsky. Such surfaces are in one-to-one correspondence with the solutions of a certain non-linear wave-equation (the so-called Sine-Gordon Equation, or SGE) that also arises in high-energy physics. SGE is an equation of soliton type and the Breather surface corresponds to a time-periodic 2-soliton solution. See more pseudospherical surfaces on the 3D-XplorMath Gallery.

--- Richard Palais (Univ. of California at Irvine, Irvine, CA)